 Welcome back everyone to our lecture series Math 12-10, Calculus 1 for students at Southern Utah University. As usual, I'm your professor today, Dr. Andrew Misalign. Great to have you. This lecture part is entitled Anti-Drivetives and is based upon Section 4.9 from Jane Stewart's Calculus textbook. And basically, we're going to undo everything we've ever done this entire semester. What do we mean by that? Well, up until this point, we've basically developed every dedicated every moment of this class getting towards the derivative. We studied limits so we can understand what is a derivative. It's a limit of a difference quotient. We've learned a lot about computing derivatives. We've talked about story problems and applications that involve derivatives. Well, as the name suggests, an anti-derivative is the opposite of a derivative. Now, before we go into the definition, let's talk a little bit about why anyone even cares about what this thing is. So first of all, functions are used in applications and we've used them all over the place, functions, functions, functions. And they've provided a lot of information to us about the total amount of a quantity. So we've looked at like cost functions, revenues functions, profit functions, distance functions, just to name a few. And derivatives of these functions provide information about the rate of change of these quantities and allows us to answer important questions on extrema, related rates, and some other things that we've talked about in this chapter and previous chapters as well. It's not always possible to find a ready-made function though that provides information about the total amount of a quantity. Sometimes it's actually more possible or more likely to collect enough information so that we can come up with a function about the rate of change of the quantity. That is, in practice, we don't always have the function y equals f of x sitting on a silver platter for us to use. Instead, we have information about f prime and we have to build the function f from that. So for example, let's suppose there's an asteroid that's flying throughout our space and we're concerned if it's going to come hit our tiny little blue marble, which we like to call home right here, right? Well, it's not known initially what the path of the asteroid is. We don't have a function for its position function apiary. But what we can do is, using astronomical methods, we can measure the speed and the trajectory, a.k.a. the velocity of the asteroid. And we can use this to construct a position function for the asteroid. And so if we construct the position function, then we can decide if that asteroid is going to hit Earth or not. And if it is, we can give Bruce Willis a call and things will be great. All right? So this idea of creating a function from the derivative is known as the anti-differentiation problem. That is, this idea of anti-derivatives. All right, so what's the formal definition? If we have a function f of x, let me take that back. So we have this function capital f of x. And if its derivative is equal to little f of x, we call capital f of x the anti-derivative of little f of x. So an anti-derivative of a function is a function whose derivative is the given function. And so a quick example of that, let's take the function capital f of x equal 10x. Then we know that if we take the derivative of this thing, well, that will just be 10. So if we take little f of x to be the constant function 10, then this tells us that it's anti-derivative, capital f of x would equal 10x. And so this is an anti-derivative for the function little f of x. And so we're reversing the derivative process that we saw earlier. As another example, if you take capital f of x this time to be x squared, then we could take, well, its derivative would then be 2x, which we'll call that little f of x. All right, little f of x. And so we have the relationship here that if little f of x equals 2x, then again, capital f is an anti-derivative of little f. Give me some more room here on the screen. So this is an anti-derivative because its derivative is little f. But I want to make mention, though, if we take a different function, g of x equals x squared plus 5. This function g of x also has the relationship that its derivative is equal to 2x, which is little f of x. So this shows us that g is also an anti-derivative. Huh, that's interesting. I wonder if I could look at another one, or if I could find another one. Well, let's look at the time right now. According to my watch, it is 803 right now. And so I'm going to take a function h of x, which looks like x squared plus 803. Well, what is its derivative? h prime is going to equal 2x again, which is again f of x. And so we get that h is also an anti-derivative of this function, little f of x. And so, like we see in that previous example, it's important to know that even though a function can have... Well, I guess I should say that it's very possible that a function can have multiple anti-derivatives. And this is actually quite commonplace. Like we saw that, you can pick whatever constant you want. And because the derivative of a constant is equal to zero, adding a constant to a function doesn't change its anti-derivative whatsoever. And so functions will have multiple anti-derivatives. But what we can rest assured about is that different anti-derivatives will only differ by their y-intercepts. That is, any two anti-derivative of the same function are really just vertical translations of each other. Or stated more formally as a theorem right here. If f of x and g of x are both anti-derivatives of the same function, little f of x on some interval, then there's a constant such that f of x minus g of x equals c. Or as we often write this, f of x equals g of x plus c. That is, these functions only differ by a constant. They differ by a constant. And the proof of this fact right here actually follows from the mean value theorem, which we saw earlier this semester. The language is a little bit different here because we didn't use the language of anti-derivatives yet. But we did prove using the mean value theorem earlier that if two functions have the same derivative, they differ only by constant. That's exactly the scenario we're in right now. So in terms of a little bit more vocabulary here, we are going to introduce a symbol. This is referred to as the integral symbol right here, the integral of f of x dx. And the integral is going to represent the family of all anti-derivatives of this function. And so you're going to see a part of the anatomy of an integral. There's this little symbol right here. This is known as the integral sign. It looks like a really elongated s, and it actually is an s. It stands for sum. We'll talk about what sums have to do with anything in a future lecture. This actually explains why there's a picture of a violin on the cover of James Stewart's calculus textbook. Because James Stewart made the observation that the f-note, the f-hole of a violin looks like an integral symbol, right? Then there's the function we're integrating. That is, this is the function we're going to look for an anti-derivative of. We call this the integrand. And then the last part, we get this little piece, the dx here. This is called the differential. The differential, amongst other things, keeps track of which variable are we going to integrate. Much in the same way that when you have the expression dy dx, the y on top is the function we will take the derivative of. The dy on the bottom, or the dx on the bottom, excuse me, tells me which variable we're going to take the derivative of. And if you put all this together, you get what's called an indefinite integral. Definite integrals we'll talk about later, but not for right now. And so using this integral notation, if capital F prime of x equals little f of x, then we get that the integral of f of x equals capital F of x plus a constant. And so you show you how this would work. If you look at the integral of 2x dx, what we're doing is we're looking for the family of anti-derivatives. We want a function whose derivatives 2x, we saw that earlier as x squared. But we always have to add in this arbitrary constant, this plus C, because we don't know what the y intercept is going to be for this function. So we have to leave it arbitrary. And so this gives us just a quick introduction to what an integral is and what some notation is for integrals. And we'll do some calculations of integrals next time.